Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation

The crossing number of graphG is theminimum number of edges crossing in any drawing ofG in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of G. We consider a conflict graph G󸀠 of G. Then, instead of minimizing the crossing number ofG, we show that it is equivalent to maximize the weight of a cut ofG󸀠. We formulate the original problem into theMAXCUT problem.We consider a semidefinite relaxation of theMAXCUT problem. An example of a case where G is hypercube is explicitly shown to obtain an upper bound.The numerical results confirm the effectiveness of the approximation.


Introduction
Let  be a simple connected graph with a vertex-set () = {V 1 , V 2 , V 3 , . . ., V  } and an edge-set () = { 1 ,  2 ,  3 , . . .,   }.The crossing number of graph , denoted cr(), is the minimum number of pairwise intersections of edge crossing on the plane drawing of graph .Clearly, cr() = 0 if and only if  is planar.It is known that the exact crossing numbers of any graphs are very difficult to compute.In 1973, Erdös and Guy [1] wrote, "Almost all questions that one can ask about crossing numbers remain unsolved."In fact, Garey and Johnson [2] prove that computing the crossing number is NPcomplete.
A 2-page drawing of  is a representation of  on the plane such that its vertices are placed on a straight horizontal line  according to fixed vertex ordering and its edges are drawn as a semicircle above or below  but never cross .

It was declared by Eggleton and Guy
Then, in 1973, Erdös and Guy [1] conjectured equality in (1).In 1993, a lower bound of cr(  ) was proved by Sýkora and Vrt' o [4]: In 2008, Faria et al. [5] constructed a new drawing of   in the plane which led to the conjectured number of crossings To the best of our knowledge, the fixed linear crossing number for   has not been established.In this paper, we discuss a method to obtain an approximation for fixed linear crossing number for hypercube graph.

2-Page Drawings of Hypercube Graph 𝑄 𝑛
Throughout this paper, we consider the ordering of hypercube graph   .Since   is defined recursively as  −1 × 2 , for Figure 1: -Cube graphs with fixed vertex ordering for  = 1, 2, 3.  = 2, . .., where  1 is a simple graph with 2 vertices together with a single edge incident to both vertices,   has 2 copies of  −1 with edges connecting between them.Given a fixed ordering on  −1 , the vertices of the first  −1 are labeled 1, 2, 3, . . ., 2 −1 and the vertices of the second  −1 are labeled 2 −1 +1, 2 −1 +2, . . ., 2 −1 +2 −1 = 2  .The two vertices between the first  −1 and the second  −1 are adjacent if and only if the sum of the labeled is 2  + 1. Figures 1 and 2 present the ordering of  1 ,  2 ,  3 and  4 which we consider throughout this paper.Notice that our method is independent on vertex ordering; therefore, for a fixed , we can apply the method (2  )! times so as to obtain the 2-page linear crossing number.
The 2-page drawing of   can be represented by drawing the vertices of   on a straight horizontal line  according a fixed vertex ordering.Each edge fully contained one of the two half-planes (pages) as a semicircle and never cross .Notice that no edge crosses itself, no adjacent edges cross each other, no two edges cross more than once, and no three edges cross in a point.For a given 2-page drawing of   with the fixed vertex ordering, a pair of edges   = (V  , V  ) and   = (V  , V  ) are potential crossing if   and   cross each other when routed on the same side of .Clearly,   and   are potential crossing if and only if Next we give the definition of conflict graph   of graph .
Definition 1.Given a graph .We define an associated conflict graph   = (  ,   ) of a graph  = (, ).There is corresponding one-to-one and onto mapping between the set of   (  ) and ().Two vertices of   are adjacent if any two edges in  are potential crossing.
For example, according to the given fixed vertex ordering of  3 (see Figure 3), are adjacent in   3 because  24 and  35 are potential crossing in a 2-page drawing of  3 .A fixed vertex ordering of  4 and its potential crossing can be seen in Figure 4.
In this paper, we are interested only in fixed linear embeddings of   .There is a crossing between   and   if and only if   and   are potential crossing and embedded on the same side of .We can see that the number of edge crossings depends on the order of vertices and on the sides to which the edges are assigned.
The 2-page linear crossing number of   , denoted by ] 2 (  ), is the minimum number of pairwise intersections of edges crossings determined by a 2-page drawing of   .The 2-page fixed linear crossing number of   is the minimum number of pairwise intersections of edges crossings determined by a 2-page drawing of   with fixed vertex ordering of

Reduction to MAXCUT Problem
In this section, we show that the problem can be reduced to the maximum cut problem.Next, we reduce the fixed linear crossing number problem to the maximum cut problem (MAXCUT).The MAXCUT problem is as follows.

Maximum Cut Problem (MAXCUT).
Given an undirected graph   = (  ,   ) the edge   of the graph is associated with nonnegative weights   .The problem is to find a cut of the largest possible weight, that is, to partition the set of   into disjoint sets  1 and  2 such that the total weight of all edges linking  1 and  2 (i.e., with one incident node in  1 and the other one in  2 ) is as large as possible.
In the MAXCUT problem, we may assume that the weights   =   ≥ 0 are defined for every pair ,  of indices: it suffices to set   = 0 for pairs ,  of nonadjacent nodes.For the unweighted graph, we assume that   = 1 for ,  = 1, 2, . . ., .
Let  be a graph with a fixed vertex permutation.Given a vertex partition ( 1 ,  2 ) of its conflict graph   , the associated cut embedding is the fixed linear embedding of  where edges corresponding to  1 and  2 are embedded to the half spaces above and below the vertex line, respectively.Lemma 2 (see [6]).
where |  | is a number of potential crossing of 2-page drawing of , which is the number of edges of   .MC(  ) is the size of the maxcut of   .
Proof.Given a 2-page (circle) drawing of , define  ⊂   as the chords that are drawn inside the circle.The edges of   with precisely one endpoint in  now correspond to edges of  that do not cross in the drawing.
Theorem 3 (see [7]).Consider a partition ( Theorem 3 reduces the fixed linear crossing number problem to the maximum cut problem (MAXCUT).In the next section, we describe the relaxation of the MAXCUT problem which leads to semidefinite programming.

Formulating MAXCUT by Semidefinite Relaxation.
In this section, we show that 2-page crossing number of hypercube graph problem can be obtained by computing a semidefinite relaxation of MAXCUT.
First of all, we introduce the adjacency matrix of  denoted Adj() as we know it is an  ×  matrix with the property From Adj() we construct the conflict graph of  denoted   .Finally, we perform MAXCUT on graph   .We use semidefinite relaxation to approximate the optimal value solution to the MAXCUT problem.Obviously the approximation is larger than the actual MAXCUT optimal value.The feasibility of the relaxation set is strictly larger than the original ones.
According to [2], the MAXCUT problem can be formulated as follows: Figure 5: The adjacency matrix of size 2 6 × 2 6 of  6 .
We call the optimal value of (6) as "OPT."Then, the relaxation of ( 6) can be rewritten as where  = [  ] is an adjacency matrix of   and  = [  ] is a feasible solution to the semidefinite relaxation.The problem ( 7) is equivalent to where  is a given adjacency matrix of   and  = [  ] is a feasible solution to the semidefinite relaxation.We call the optimal value of (8) as "SDP." As we have seen from the relation (4), we let |   | be the number of potential crossing of 2-page drawing of   with our fixed vertex ordering (i.e., |   | is the number of edges of    ).We can determine |   | by considering the upper half of the main diagonal of the adjacency matrix of   .Definition 4. Let  = [  ] be the  ×  adjacency matrix of .The element   where  +  =  + 1 is called minor diagonal of adjacency matrix of  and the element   where  +  =  + 1,  <  is called semiminor diagonal of adjacency matrix of , denoted by smd().
For simplicity, the size of smd() is a number of elements in smd().Let   be the adjacency matrix of graph   .Therefore   is 2  × 2  symmetric matrix.It is clear that the size of smd(  ) is 2 −1 .
Let  1 and  2 be adjacency matrices of graphs  1 and  2 , respectively; we say that the number of potential crossing between  1 and  2 , denoted by PC[ 1 ,  2 ], is simply the number of potential crossing between 2-page drawing of graph  1 and  2 .The adjacency matrix of size 2 6 × 2 6 of  6 with respect to our ordering is presented in Figure 5.
Lemma 6.For every adjacency matrix of   ,   , where  ≥ 4, there exists adjacency matrix of  3 ,  3 , which is a submatrix embedding in   .The number of submatrix  3 embedding in   is 2 −3 .
Lemmas 5-9 follow directly from the definitions.
Proof.We prove this lemma by considering the number of potential crossing of 2-page drawing of   with our fixed vertex ordering.Since   has 2 copies of  −1 with some edges connecting between them, the number of potential crossing of   is a result of twice of the number of potential crossing within  −1 together with PC[ 3 , smd(  )] and PC[smd(  ), smd(  )],  = 4, 5, . . .,  − 1.
Hence, it is enough to show that the number of potential crossing between 2-page drawing of  −1 and  −1 is equal to (), where Note () is the number of potential crossing between all of submatrices  3 and smd(  ) and also between smd(Q  ) and smd(  ),  = 4, 5, . . .,  − 1.By Lemmas 6 and 9, We precede by mathematical induction on .For  = 5, it can be easily seen that (5) = 176 by counting.Assuming (15) holds true, now we consider ( + 1) as a number of potential crossing between all of the submatrices  3 and smd( +1 ) and also between smd(  ) and smd( +1 ),  = 4, 5, . . ., .By the Lemmas 5, 6, 7, 9, 8, and 10, The next theorem shows how effective the relaxation is.
Theorem 12 (see [8]).Let OPT be the optimal value of the MAXCUT problem and SDP be the optimal value of the semidefinite relaxation.Then Theorem 12 guarantees that the optimal value of the MAXCUT is close to the optimal value of the semidefinite relaxation.From (4), we have where |   | is a number of potential crossing of 2-page drawing of   .AP(] 2 (  )) is an approximation of 2-page fixed linear crossing number of   and AP(MC(   )) is an approximation of MC(   ).

Experimental Results.
In this section, we consider the hypercube graph   for  = 4,5,6.Then, we give some examples for approximating the problems of the semidefinite relaxation in the form (8). We approximate this problem via MATLAB program together with an optimization toolbox called "SeDuMi."The SeDuMi is a package for solving optimization problems with linear, quadratic, and semidefinite constraints.
In Table 1, the second column shows numerical results for the approximation of the MAXCUT on the associated conflict graph    by using the semidefinite relaxation.It is well known that this problem can be solved in a polynomial time.The third column displays the numbers of potential crossing of 2page drawing of   referring to our fixed vertex ordering that we evaluate from ( 14).Notice that this potential crossing of 2page drawing of   is the exact value.From (19), we calculate the approximation of 2-page fixed linear crossing number of   for  = 4, 5, 6.The results are shown in the last column.

Figure 3 :
Figure 3: The 2-page drawing of  3 with fixed vertex ordering.

Figure 4 :
Figure 4: The 2-page drawing of  4 with fixed vertex ordering.
Let   be the set of edges in   with one endpoint in  1 and one endpoint in  2 , that is, the cut given by ( 1 ,  2 ).By definition of   , we know that every crossing in the cut embedding associated with ( 1 ,  2 ) corresponds to an edge in   such that either both its endpoint belong to  1 or both belong to  2 , that is, to an edge in   \   .Thus, the number of crossings is |  | − |  |.As |  | is constant for a fixed vertex permutation, the result follows.
1 ,  2 ) of   .Then the corresponding cut embedding is a fixed linear embedding of  with a minimum number of crossings if and only if ( 1 ,  2 ) is a maximum cut in   .Proof.